Comparing the efficiency of numerical techniques for the integration of variational equations

Geometrical properties of local dynamics in Hamiltonian systems: The Generalized Alignment Index (GALI) method

Application of the SALI chaos detection method to accelerator mappings

In this Chapter, we discuss in a unified way equilibrium points, periodic orbits and their stability, which constitute local concepts of Hamiltonian dynamics together with ordered and chaotic motion, which are the concern of a more global type of analysis. Using the Fermi Pasta Ulam β (FPU − β) model as an example, we study the destabilization properties of some of its simple periodic orbits and connect them to wider aspects of the motion in phase space. We show numerically that the spectrum of positive Lyapunov exponents, characterizing a region of strong chaos, becomes invariant in the thermodynamic limit, verifying that the Kolmogorov-Sinai entropy, defined as the sum of these exponents, is an extensive property of the system. The Chapter ends with an introduction of the smaller alignment index (SALI) and the generalized alignment index (GALI) criteria for distinguishing ordered from chaotic motion.

Understanding the regular or chaotic nature of orbits in galaxies is undoubtedly an issue of great importance. We determine the character of orbits of stars moving in the meridional plane (R, z) of an axially symmetric time-independent galactic model with a spherical central nucleus, and a flat biaxial oblate dark matter halo component. In particular, we try to reveal the influence of the fractional portion of dark matter on the structure and also on the different families of orbits of the galaxy, by monitoring how the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families evolve when the ratio of dark matter to luminous mass varies. The smaller alignment index (SALI) was computed by numerically integrating the equations of motion as well as the variational equations to extensive samples of orbits in order to distinguish between ordered and chaotic motion. In addition, a method based on the concept of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used to identify the various families of regular orbits and also to recognize the secondary resonances that bifurcate from them. The investigation is carried out both in the physical (R, z) and the phase \((R,\dot {R})\) space for a better understanding of the orbital properties of the system. The numerical computations reveal that in both cases, the fractional portion of dark matter influences more or less, the overall orbital structure of the system. It was observed however, that the evolution of the percentages of all types of orbits as a function of the fractional portion of dark matter strongly depends on the particular type of space (physical or phase) in which the initial conditions of orbits are launched. The results are compared with the similar earlier work.

We consider the dynamical behavior of fermionic instanton solutions of the Thirring Model with two forcing terms. In particular, the study focuses on the effect of the frequency and amplitude of the forcing terms on the behavior of fermionic instanton solutions. Numerical analysis based on the Smaller Alignment Index (SALI) method, Bifurcation Diagram and Permutation Entropy (PE) are used to show how and which dynamical behaviors occur in the system. Color maps and diagrams of the SALI time (St) (the time St required for the SALI index to below a threshold value of 10−12) and PE of the system with respect to varying of frequency and amplitude values of the forcing terms are plotted comparatively to determine different dynamical behavior of the system. The study shows that fermionic instanton solutions exhibit a wide variety of dynamical behavior due to two forcing terms. Furthermore, it is emphasized that SALI time (St) can be easily compared with bifurcation diagram and complexity method as a fast, efficient and precise method to investigate different degrees of chaos. In general, the instanton solutions with two forcing terms have been observed to exhibit different type of dynamical behavior. In this study, identical and symmetric coexisting attractors are demonstrated for these different types of behavior of the forced Thirring instanton.

The scope of this work is to perform a numerical investigation of the orbital dynamics for a test particle in the pseudo-Newtonian Hill problem. Large two-dimensional sets of initial conditions of prograde and retrograde orbits are investigated. The orbits are classified as bounded (chaotic, sticky or regular), escaping and collision orbits. The smaller alignment index (SALI) method is used to identify chaotic orbits. Additionally, the influence of the energy (or equivalently the value of the Jacobi constant) and of the Schwarzschild radius on the orbital structure of the system are determined. Our numerical results are compared with related previous ones, corresponding to the classical version of the Hill problem.

Quantitative orbit classification of the planar restricted three-body problem with application to the motion of a satellite around Jupiter

Observational evidence strongly supports the existence of a Super Massive Black Hole at the Galactic center, surrounded by dense stellar clusters. Modeling galactic centers with intricate structures like shells and rings pose challenges, prompting the use of simplified models such as a spherical monopole potential with a multipolar halo mass distribution. This approach, employing a multipolar expansion model, provides versatility for numerical analyses, revealing the complex dynamics of stars in this region. Pseudopotentials like Paczynsky-Wiita and Artemova-Bjornsson-Novikov are utilized to simulate the impacts of strong gravity from nonrotating and rotating compact objects respectively, elucidating their influence on stellar dynamics. Chaos naturally arises due to noncentral forces, visualized using the Poincaré sectiontechnique. Of particular importance is the utilization of the Smaller Alignment Index (SALI), a powerful nonlinear dynamical tool, which categorizes particle orbits as escaping, regular, sticky, or chaotic. We exhaustively examine all combinations of multipolar moments up to the octupolar term along with spin using this tool, which had not been studied earlier. SALI provides a straightforward yet efficient method for assessing the interplay between the system's different multipolar moments, their combinations, and spin. Thus, our findings offer insights into the dynamics of compact objects enshrouded in a halo mass distribution and lay the groundwork for understanding complex astrophysical systems in galactic centers.

The escape dynamics of the stars in a barred galaxy composed of a spherically symmetric central nucleus, a bar, a flat thin disk and a dark matter halo component is investigated by using a realistic three degrees of freedom (3-dof) dynamical model. Modern colour-coded diagrams are used for distinguishing between bounded and escaping motion. In addition, the smaller alignment index (SALI) method is deployed for determining the regular, sticky or chaotic nature of bounded orbits. We reveal the basins of escape corresponding to the escape through the two symmetrical escape channels around the Lagrange points $L_2$ and $L_3$ and also we relate them with the corresponding distribution of the escape times of the orbits. Furthermore, we demonstrate how the stable manifolds, around the index-1 saddle points, accurately define the fractal basin boundaries observed in the colour-coded diagrams. The development scenario of the fundamental vertical Lyapunov periodic orbit is thoroughly explored for obtaining a more complete view of the unfolding of the singular behaviour of the dynamics at the cusp values of the parameters. Finally, we examine how the combination of the most important parameters of the bar (such as the semi-major axis and the angular velocity) influences the observed stellar structures (rings and spirals) which are formed by escaping stars guided by the invariant manifolds near the saddle points.

We provide a concise presentation of the Smaller (SALI) and the Generalized Alignment Index (GALI) methods of chaos detection. These are efficient chaos indicators based on the evolution of two or more, initially distinct, deviation vectors from the studied orbit. After explaining the motivation behind the introduction of these indices, we sum up the behaviors they exhibit for regular and chaotic motion, as well as for stable and unstable periodic orbits, focusing mainly on finite-dimensional conservative systems: autonomous Hamiltonian models and symplectic maps. We emphasize the advantages of these methods in studying the global dynamics of a system, as well as their ability to identify regular motion on low dimensional tori. Finally we discuss several applications of these indices to problems originating from different scientific fields like celestial mechanics, galactic dynamics, accelerator physics and condensed matter physics.

Study on the finite element method of Hamiltonian system with chaos

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of [Maffione et al., 2011b] for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of [Maffione et al., 2011b] for mappings to the 2D [Hénon & Heiles, 1964] potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba & Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince & Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stöer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We first show that a combination of the FLI/OFLI, the MEGNO and the GALI 2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stöer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stöer.

We study some new universal aspects of diffusion in chaotic systems, especially such having very large Lyapunov coefficients on the chaotic (indecomposable, topologically transitive) component. We do this by discretizing the chaotic component on the Surface-of-Section in a (large) number $N$ of simplectically equally big cells (in the sense of equal relative invariant ergodic measure, normalized so that the total measure of the chaotic component is unity). By iterating the transition of the chaotic orbit through SOS, where $j$ counts the number of iteration (discrete time), and assuming complete lack of correlations even between consecutive crossings (which can be justified due to the very large Lyapunov exponents), we show the universal approach of the relative measure of the occupied cells, denoted by $\rho(j)$, to the asymptotic value of unity, in the following way: $\rho(j) = 1 - (1-\frac{1}{N})^j$, so that in the limit of big $N$, $N\to \infty$, we have, for $j/N$ fixed, the exponential law $\rho(j) \approx 1 - \exp (-j/N)$. This analytic result is verified numerically in a variety of specific systems: For a plane billiard (Robnik 1983, $\lambda=0.375$), for a 3-D billiard (Prosen 1997, $a=-1/5, b=-12/5$), for ergodic logistic map (tent map), for standard map ($k=400$) and for hydrogen atom in strong magnetic field ($\epsilon=-0.05$) the agreement is almost perfect (except, in the latter two systems, for some long-time deviations on very small scale), but for H\'enon-Heiles system ($E=1/6$) and for the standard map ($k=3$) the deviations are noticed although they are not very big (only about 1%). We have tested the random number generators (Press et al 1986), and confirmed that some are almost perfect (ran0 and ran3), whilst two of them (ran1 and ran2) exhibit big deviations.

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps.

In this work, we try to shed some light to the nature of orbits in a three-dimensional (3D) potential of a perturbed harmonic oscillator with eight possible channels of escape, which was chosen as an interesting example of open 3D Hamiltonian systems. In particular, we conduct a thorough numerical investigation distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering unbounded motion for several values of the energy. In an attempt to discriminate safely and with certainty between ordered and chaotic motion, we use the Smaller ALingment Index (SALI) detector, computed by integrating numerically the basic equations of motion as well as the variational equations. Of particular interest is to locate the basins of escape toward the different escape channels and connect them with the corresponding escape periods of the orbits. We split our study into three different cases depending on the initial value of the \(z\) coordinate which was used for launching the test particles. We found that when the orbits are started very close to the primary \((x,y)\) plane the respective grids exhibit a high degree of fractalization, while on the other hand for orbits with relatively high values of \(z_0\) several well-formed basins of escape emerge thus reducing significantly the fractalization of the grids. It was also observed that for values of energy very close to the escape energy the escape times of orbits are large, while for energy levels much higher than the escape energy the vast majority of orbits escape extremely fast or even immediately to infinity. We hope our outcomes to be useful for a further understanding of the escape process in open 3D Hamiltonian systems.